Classifications of Several Classes of Armendariz-like Rings Relative to an Abelian Monoid and Its Applications

Jianwei He () and Yajun Ma
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Jianwei He: School of Mathematics, Nanjing University, Nanjing 210093, China
Yajun Ma: School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, China

Mathematics, 2025, vol. 13, issue 5, 1-22

Abstract: Let M be an Abelian monoid. A necessary and sufficient condition for the class A r m M of all Armendariz rings relative to M to coincide with the class A r m of all Armendariz rings is given. As a consequence, we prove that A r m M has exactly three cases: the empty set, A r m , and the class of all rings. If N is an Abelian monoid, then we prove that A r m M × N = A r m M ⋂ A r m N , which gives a partial affirmative answer to the open question of Liu in 2005 (whether R is M × N -Armendariz if R is M -Armendariz and N -Armendariz). We also show that the other Armendariz-like rings relative to an Abelian monoid, such as M -quasi-Armendariz rings, skew M -Armendariz rings, weak M -Armendariz rings, M - π -Armendariz rings, nil M -Armendariz rings, upper nil M -Armendariz rings and lower nil M -Armendariz rings can be handled similarly. Some conclusions on these classes have, therefore, been generalized using these classifications.

Keywords: Armendariz rings; M -Armendariz rings; Armendariz-like rings; Armendariz-like rings relative to an Abelian monoid (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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